Mine Sweeper Advantage Manual

“Beat those suckers quicker and faster!”

Strategy

Minesweeper strategy is the art of solving games. Techniques

include learning patterns and where to click first, using guessing

tactics and developing efficient clicking and mouse movement.

Patterns

A pattern is a common arrangement of numbers that has only one solution. If

you memorize a pattern it will reduce the amount of time you waste thinking.

Before you start learning patterns, you should learn the basics. If a number is

touching the same number of squares, then the squares are all mines. Here are

some examples:

There are two basic patterns which combine to make all other patterns. The first

is 1-1 and the second is 1-2. Whenever you see a 1-1 pattern starting from an

edge (or where an opened square functions as an edge) the 3rd square over is

empty. This makes sense because the first 1 touches two squares which must

contain the mine, while the second 1 touches a third square, which must be

empty. Whenever you see a 1-2 pattern the 3rd square over is always a mine. This

makes sense because the first 1 touches two squares which must contain the

mine, while the 2 touches a third square, which must contain the second mine.

Here are some examples:

The two most famous patterns are 1-2-1 and 1-2-2-1. These are so common new

players should memorize them immediately. If you look carefully they are just

combinations of the 1-2 pattern.

At first it seems like there are many patterns. If you study them, they are actually

1-2-1 and 1-2-2-1 patterns or combinations. These in turn are variations of the

basic 1-2 pattern. Each set of numbers reduces when you subtract known mines.

Here are some final examples:

Guessing

Sometimes in Minesweeper you have to guess. A typical case is a 50/50 situation

where one mine is hidden in two squares. Guess quickly and move on. Thinking

does not improve your chance of guessing correctly and wastes time. Waiting to

see if you guessed right also wastes time.

You will know soon enough if you are wrong! Do not delay taking these forced

guesses - solving the rest of the board first is a waste of time unless you are

going to guess correctly.

Many players get impatient and guess instead of solving. Do not guess unless it

is necessary. The fastest way to solve 'Example A' is to click the unopened

squares in a row. But if you click fast there is no time to react, so you will lose if

the middle square is a mine. You have just guessed for no reason!

A smart player will click the outer two squares first, which allows enough time to

react to the initial click and decide if there is a mine.

Opening safe squares is as important as finding mines. If you can prove a square

is safe, open it instead of guessing where the mine is. In 'Example B' there is a

mine in the two yellow squares.

Instead of guessing, open the safe 3rd square. This can allow you to open even

more squares (marked blue) which may help you solve the original guess.

Often you can improve your chance of guessing right. There might be an

arrangement of numbers with more than one solution, and the solutions require

different amounts of mines. Instead of guessing, you can solve it by flagging the

rest of the board and seeing how many mines are left.

If you insist on guessing, think about the mine density of the level you are

playing. For example, the solution with more mines is more likely on Expert than

on Intermediate. Still keep in mind the density of each level is pretty low, so less

dense solutions are more common overall.

Perhaps you have solved part of a board and need to guess in order to reach the

rest of the board. You can improve your chance of surviving by clicking randomly!

The average chance of hitting a mine is 0.206 on Expert and 0.156 on

Intermediate and Beginner.

These odds are much better than a 50/50 guess. Remember you are more likely

to get openings by clicking on edges. Your bravery is often rewarded by finding

that the original 'guess' becomes solvable when approached from a different

direction.

Another important thing to remember is usefulness. If two solutions are equally

likely, choose the one that will help most if it is correct. Sometimes one solution

prevents another guess or creates an easier arrangement of mines.

When a guess has more empty squares than mines involved, it is always better to

guess an empty square instead of guessing the mine. Flaggers often make the

mistake of guessing the mine because they love to chord. Another mistake is to

choose an empty square that creates a new guessing situation, such as turning a

33/66 problem into a 50/50 problem. Guess the empty square that solves the

original problem.

Always choose the most likely solution. This can be very difficult to calculate!

Local probability is easy to calculate but is usually wrong. For example, in the

image below some squares are both 50/50 and 66/33 guesses!

When all unsolved areas are considered, a simple 50/50 guess often has one

square much more likely to contain the mine. A general rule of thumb is that if

one square in a 50/50 situation touches a high number, it is more likely to be a

mine than the other square.

The following example illustrates many of the above points. It looks like there are

three unavoidable 50/50 guesses, and two unavoidable 66/33 guesses. One

strategy is to guess quickly and hope for the best. This option will give the best

score if you survive.

A second strategy is to click a random square that does not touch any numbers.

This usually has better odds of being safe and often helps solve the game.

A third strategy is to determine the number of mines remaining by flagging the

rest of the board. This reduces the number of solutions. In this example there are

79 possible solutions but only 2 contain 4 mines.

A fourth strategy is to guess in the most useful place. Clicking square I has the

potential to eliminate all the other guesses! For example, if it is a 4 or 7 the

game can be solved no matter how many mines remain.

A fifth strategy is to guess the most likely solution. A mine is more likely in L

than K and more likely in H than D. A final strategy is to calculate the exact

probability of each square taking the entire game into consideration. This is the

hardest but most accurate method. Results for this example are available.

The guessing strategy you choose depends on whether you want to win more

games or only fast games. Some strategies improve your chance of winning but

are time consuming. Others are fast but risky.

First Click

The first click in Minesweeper is always safe, but where is the best place to start?

It depends whether you want quantity or quality.

Your best chance of finding an opening is in a corner, then on an edge, then in

the middle. Emmanuel Brunelliere (France) calculated the theoretical odds as

follows:

Beginner Intermediate Expert

Corner 59.54% 59.94% 49.94%

Edge 42.14% 42.61% 31.42%

Middle 25.09% 25.54% 15.69%

The first click is always safe because any mine is moved to the top left corner or

nearest empty square to its right. This means the top left corner gives fewer

openings than the other corners. It also means fewer openings result from the

edge and middle squares touching the top left corner. Exact values for each

square are on his website. Most of the variation is due to low outlier values near

the top left corner.

Beginner Intermediate Expert

Corner 50 - 60 % 50 - 60 % 40 - 50 %

Edge 34 - 42 % 36 - 43 % 25 - 32 %

Middle 19 - 24 % 21 - 26 % 12 - 16 %

Your best chance of getting a large opening is in the middle, then on an edge,

then in a corner. So far no one has calculated the theoretical advantage, but Tim

collected actual results from Windows Minesweeper.

The biggest openings occur in the very center of the board and decrease as you

approach edges. The biggest openings from clicking on an edge are in the

middle and decrease as you approach corners. This chart shows the variations in

the average number of squares for each opening:

Beginner Intermediate Expert

Corner 18 27 16

Edge 20 - 24 31 - 42 19 - 26

Middle 23 - 32 35 - 66 23 - 41

In summary, the best place to start depends on your preference for size or

frequency. Large openings are more helpful but you will lose more games trying

to find them. Small openings can be difficult but you will start more games. It is

possible the benefits of either method cancel each other.

Efficiency

The fewer clicks you take, the faster you will finish. Learn to be efficient.

The game ends when all safe squares are open, not when all mines are flagged.

Beginners often waste time flagging every mine. The only good reason to flag is

to clear more squares by chording. So before you place a flag, decide if it is

useful.

Some players never flag because time spent placing flags could be better used to

open more squares. This style is called No Flags, or NF. Flaggers argue that flags

allow you to chord and clear multiple squares at the same time.

It is generally agreed that NF is more efficient near high numbers (5,6,7,8) while

Flagging is more efficient near low numbers (1,2,3,4). Near a high number like 7 a

NF player needs only one click to open the safe square, but a Flagger needs

seven flags and a chord.

Near a low number like 1 a Flagger would place one flag and chord, but a NF

player would need as many as seven clicks to open the safe squares. It is also

generally agreed that NF is more efficient on low 3BV boards while Flagging is

more efficient on high 3BV boards.

For example, an Intermediate game with a 3BV of 40 has an average of one

number touching each mine, while a 3BV of 120 has an average of three. A

perfect NF player would need 40 and 120 clicks. A very inefficient and unlucky

Flagger would need 80 clicks (40 flags, 40 chords) for both games.

These examples are extreme cases but show the general reasoning. In reality NF

players are not perfect and waste clicks at full speed, while Flaggers never need

to flag all mines or chord on every number. If a player uses only NF or Flagging,

there is probably no advantage to either method. The advantage comes when

both techniques are combined and the player uses the most efficient solution for

each situation.

If you flag you can save time by using the 1.5 click technique. The traditional way

is moving the right button down and up to flag and both buttons down and up

to chord. The new method moves the right button down to flag, the left button

down, and both up to chord.

This eliminates one movement from every flag and chord combination. As long

as the right button starts going down before the left button, the flag will get

placed. The shorter the gap, the more time you save. You can nearly double your

flagging speed with this method.

Here are some examples of efficient flagging:

Here are some examples of efficient NF:

It is usually easy to tell whether NF or Flagging is better in a given situation, but

here is a complicated example:

An important way to increase solving speed is to make fewer mouse movements.

It takes time to move your mouse. New players often follow their eyes with the

mouse instead of moving it intentionally toward a target.

The next stage in reducing movement is learning to 'see' the solved board. This

often allows you to solve at your current mouse location. If your mouse is near

the 2 in 'Example A' you can flag the red square and chord instantly. This is

obvious to a professional player because they have solved the adjacent squares in

their head. A new player would have to move elsewhere and come back later. The

red square in 'Example B' can be similarly solved. Less movement equals better

scores.

The best way to improve efficiency is to play slowly. Find the most efficient

solution and path to each problem before pressing any buttons. You will soon see

improvements while playing at full speed.

And of course, the best way to improve your efficiency and score is to practice,

which you can do when you download Mine Sweeper II, for free.